Yes, if not Jesus himself, then at least somebody. Odd how there is no shortage of writings after the fact.
However, everybody knows that the father of the Greek rational and critical thought, Socrates, never wrote a line of philosophy. Jesus and Socrates have often been compared. One of their common points is that we know only indirectly what they said. Others wrote their sayings down for them. Socrates preferred a “living” philosophy made of conversations with people he met in the street. He had no school and no books. For him, philosophy cannot be enclosed in formulas; it is a research made orally and in common.
Consequently, we must be aware that philosophy is not so evidently related to literacy. This is also the case for Socrates’ best pupil, Plato.
"Plato has pointed out the dangers of written works. In his Seventh Letter he states that he never himself wrote in “the sublime questions of philosophy” (341B– D) and that
no serious man will seriously write on serious problems, because he would so lay his thought open to the misunderstanding of the crowd (344B)….
Further, in the same dialogue Socrates himself remarks that a written work is a child without father— it cannot protect itself (275E)—and that
writing is deceptive like painting; the latter depicts beings that are falsely living and cannot answer questions; likewise, the former draws up books that can signify but one thing and are unable to provide explanations by themselves, shades of meaning, and so on (cf. Protagoras 329A). Moreover, the book escapes its creator’s control; it soon becomes everybody’s toy and is exposed to the danger of losing its true meaning.
Thus, for Plato, the oral discourse is better than the written one. Pp. 83-84.
Solère, Jean-Luc. “Why did Plato Write?” in Draper, Jonathan A (Ed).
Orality, Literacy, and Colonialism in Antiquity.
Leiden: Brill, 2004. p 83-91..
Q and Thomas tell us that there were those that felt the need to write down and compile a collection of sayings attributed to a Jesus by their very existence, or at least the undisputed existence of the Thomas gospel of sayings which includes much of Q. Why would it not occur at least to one or a few members of Jesus' entourage to take note, or someone out of the enormous crowds of people that formed around Jesus and followed him everywhere he went?
Because of a general feeling in antiquity that writing was inferior to oral transmission?